25-stability

25-stability - STABILITY Math 21b, O. Knill HOMEWORK: 7.6...

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Unformatted text preview: STABILITY Math 21b, O. Knill HOMEWORK: 7.6 8,20,42,38*,46*, 8.1 10,24,6*,56* Due: Wednesday or Monday after Thanksgiving. LINEAR DYNAMICAL SYSTEM. A linear map x 7 Ax defines a dynamical system . Iterating the map produces an orbit x , x 1 = Ax, x 2 = A 2 = AAx, ... . The vector x n = A n x describes the situation of the system at time n . Where does x n go when time evolves? Can one describe what happens asymptotically when time n goes to infinity? In the case of the Fibonacci sequence x n which gives the number of rabbits in a rabbit population at time n , the population grows essentially exponentially. Such a behavior would be called unstable . On the other hand, if A is a rotation, then A n ~v stays bounded which is a type of stability . If A is a dilation with a dilation factor < 1, then A n ~v 0 for all ~v , a thing which we will call asymptotic stability . The next pictures show experiments with some orbits A n ~v with different matrices. . 99- 1 1 stable (not asymptotic) . 54- 1 . 95 asymptotic stable . 99- 1 . 99 asymptotic stable . 54- 1 1 . 01 unstable 2 . 5- 1 1 unstable 1 . 1 1 unstable ASYMPTOTIC STABILITY. The origin ~ 0 is invariant under a linear map T ( ~x ) = A~x . It is called asymptot- ically stable if A n ( ~x ) ~ 0 for all ~x IR n . EXAMPLE. Let A = p- q q p be a dilation rotation matrix. Because multiplication wich such a matrix is analogue to the multiplication with a complex number z = p + iq , the matrix A n corresponds to a multiplication with ( p + iq ) n . Since | ( p + iq ) | n = | p + iq | n , the origin is asymptotically stable if and only if...
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